Instrument Science Report ACS 2000-11
Geometric Distortion Table:
IDCTAB
W. Hack, C. Cox
December 6, 2000
ABSTRACT
The new reference table, IDCTAB, will support the description of geometric distortion
models for instruments. This report describes the columns in the table and how the coefficients in the table can be used.
Introduction
Many instruments require some sort of reference file for describing the geometric distortion, yet a single format has not yet been developed that satisfies each instrument’s needs.
The description of the geometric distortion can take many different forms, ranging from
images of pixel area variations to polynomial fits. One common element shared by all the
instruments is the Science Instrument Aperture File (SIAF) which contains descriptions of
the distorted and undistorted aperture positions. The conversion between distorted and
undistorted positions in the SIAF is controlled by a polynomial fit performed by the instrument teams. This commonality provides the basis for generating a small and simple
reference file which can be used by any instrument to describe the detector’s geometric
distortion using the same fitting method.
This report describes the format of this common reference file based on the polynomial fit
used by the SIAF. The IDCTAB, as referenced by the ACS, contains the coefficients for
this fit along with other information necessary for applying to each image. The format has
been developed to be as general as possible so that instruments other than ACS can also
use it along with any software designed to work with this table.
Copyright© 1999 The Association of Universities for Research in Astronomy, Inc. All Rights Reserved.
Instrument Science Report ACS 2000-11
Distortion Model
A single model for the geometric distortion has been used for WFPC2, STIS and ACS in
the SIAF. This model ties in the aperture position to the detector pixel positions and
accounts for geometric distortion to support accurate target acquisitions, image processing, and photometry. The X and Y pixel coordinates for the reference pixel for this model
gets recorded in the science headers as CRPIX1 and CRPIX2.
The translation from detector pixel coordinates to sky position includes the scaling and
distortion correction with the origin at the reference position where the detector and sky Y
axes are assumed to be exactly parallel. The relationship between these systems is defined
for the SIAF as
k
xc =
i
∑ ∑ ai, j ( x – xr
k
) j( y
– yr
)i – j
yc =
i=0j=0
i
∑ ∑ bi, j ( x – xr ) j ( y – yr ) i – j
i=0j=0
where (x,y) are the positions from the image, (xr,yr) is the reference pixel position, (xc,yc)
are the corrected pixel positions and k is the polynomial order of the fit. The orientation of
the (xc,yc) system is normally chosen so that the corrected yc axis is parallel to the input y
axis. This relation thus converts detector coordinates in pixels to corrected position in arcseconds. The only limitation to this fit is imposed by the interface document ICD-26
which limits the SIAF polynomial order to 5. Fortunately, this limit does not prevent us
from using higher order fits in the IDCTAB should it be necessary. These polynomials
expand out to the form
x c = a 00 + a 10 y + a 11 x + a 20 y 2 + a 21 xy + a 22 x 2 + …
The values of a 00 and b 00 will always be zero, since xc=0 and yc=0 at the reference position (x,y) = (xr,yr). Also, choosing yc parallel to y forces a10 = 0.
A similar form is used for the inverse relation using coefficients c and d for the x and y fits
respectively; specifically,
k
x = xr +
i
∑∑
k
c i , j x cj y c( i – j )
y = yr +
i=0j=0
i
∑ ∑ d i, j xcj yc( i – j )
i=0j=0
Although this description appears asymmetric, the origin of each set of coefficients is the
same point. This inverse relation provides the conversion from units of arcseconds to distorted input pixels.
File Format
The IDCTAB will be stored as a FITS file with a FITS table in the first extension. The primary header will only contain basic FITS keywords along with information about the
generation of the reference file. Additional keywords in the primary header would include:
2
Instrument Science Report ACS 2000-11
INSTRUME
S
Name of the Instrument
DETECTOR
S
Name of the detector used
The first extension to the FITS file would then hold the table with the description of the
geometric distortion. Its header would contain only the basic FITS keywords necessary for
the extension plus:
NORDER
I
Order of the polynomial fit used for the distortion
PARITY
I
Parity value for conversion from (x,y) to (V2,V3)
The keyword NORDER provides a way to ascertain up front what order of polynomial has
been stored in the table, while PARITY describes the relationship between pixel coordinates and V2,V3. The table itself would then consist of the columns:
DIRECTION
S
%8s
Application direction of coefficients (FORWARD or INVERSE)
DETCHIP
S
%8s
ID of chip/detector used for observation
WAVELENGTH
R
%12.4f
XSIZE
I
%8d
Raw image size in pixels in X direction
YSIZE
I
%8d
Raw image size in pixels in Y direction
XREF
R
%12.4f
X position of reference pixel
YREF
R
%12.4f
Y position of reference pixel
XCOM
R
%12.4f
X position of common point in fit
YCOM
R
%12.4f
Y position of common point in fit
V2REF
R
%12.4f
V2 position of reference point (arcsec)
V3REF
R
%12.4f
V3 position of reference point (arcsec)
THETA
R
%12.4f
Angle from V3 axis anti-clockwise to Y axis (degrees)
SCALE
R
%12.4f
Scale of square corrected pixel in arcsec
CXSIZE
I
%8d
Corrected image size in X direction (pixels)
CYSIZE
I
%8d
Corrected image size in Y direction (pixels)
CXREF
R
%12.4f
X value of reference pixel in corrected system
CYREF
R
%12.4f
Y value of reference pixel in corrected system
CX10,CX11,...
R
%20.6e
Distortion Coefficients for X position
CY10,CY11,...
R
%20.6e
Distortion Coefficients for Y position
Central wavelength of fit
The table contains one row for each part of the detector that has its own distortion correction. For example, the ACS WFC table would contain 1 row for each 2048x4096 chip that
makes up a WFC observation where each row would have a different value of DETCHIP.
3
Instrument Science Report ACS 2000-11
An additional row for each chip may be included for the inverse fit and they would be
flagged with the DIRECTION value of ‘INVERSE’. Naturally, instruments with only one
detector (such as STIS CCD) would only have 1 row for the forward fit and an optional
row for the inverse fit. There may also be separate sets of these rows for different central
wavelengths, as determined by the optical elements used for the observation, with each set
having its own value for WAVELENGTH. The units of WAVELENGTH can vary for each
instrument since IR detectors operate in microns as opposed to Angstroms for a UV
imager.
Columns for Calibrated Values
Any distortion model relies on more than the fit to properly apply the model to the data.
The columns XSIZE and YSIZE provide the full input size of an uncorrected (distorted)
image from each detector/chip in the observation. This provides some perspective for the
location of the reference position given in the columns XREF,YREF and for the common
fit point given in the columns XCOM,YCOM. The reference position corresponds to the
nominal targeting position for the observation and serves as the zero-point for the distortion model (as discussed above with the equations). The pixel position provided by
XCOM and YCOM corresponds to the common position for the global fit. This position is
determined by taking the midpoint between each chip’s reference positions in V2,V3 and
calculating the corresponding x,y position for each chip. The fit can be applied to corrected pixel positions relative to this common point’s corrected position to result in a
global fit.
The columns V2REF and V3REF provide the telescope position for the reference position
given in XREF and YREF with THETA providing the angle between the telescope’s V3
axis counterclockwise to the images’s Y axis. The rotation from x to y is commonly in the
opposite sense to that from V2 to V3. We express this by setting the value of PARITY to
-1. The transformation from corrected image pixels (X,Y) to V2,V3 then becomes:
V2 = V2REF + SCALE ( PARITY ( x c – X ref ) cos ( THETA ) + ( y c – Y ref ) sin ( THETA ) )
V3 = V3REF + SCALE ( ( – PARITY ) ( x c – X ref ) sin ( THETA ) + ( y c – Y ref ) cos ( THETA ) )
where (Xref,Yref) is the corrected position for the reference point.
Distortion Coefficients Columns
Applying the fit to an image will result in an image where the pixels are shifted around to
correct for the distortion, but a couple of decisions remain to be made: output size and output pixel scale. Both of these are completely up to the end-user, however, this reference
file contains the columns CXSIZE, CYSIZE, and SCALE to provide calibrated default
values. The values for CXSIZE and CYSIZE specify an output size for the calibrated,
undistorted image. These values will be a value which can be calibrated and set to provide
an output image which completely contains the new image without trimming any input
4
Instrument Science Report ACS 2000-11
data. Similarly, the value for SCALE will specify a canonical, calibrated value which
would be applied to the output image in lieu of any end-user overrides. This will allow
data to be processed automatically with this reference table to produce output images of
consistent pixel scale suitable for astrometric purposes subject to the errors in the
correction.
Finally, the columns CXnn and CYnn would then contain the values for the polynomial fit
for the distortion for that chip alone. These values would convert an input pixel position to
an undistorted position assuming the value of the DIRECTION column were ‘FORWARD’. Alternatively, if DIRECTION was specified as ‘INVERSE’ for the row, the
coefficients would specify the application of the distortion to undistorted pixel positions
resulting in an distorted image.
Using the Table
The column names for the coefficients contain the indices for the coefficients from the
polynomial fit, such as CX11 corresponding to the a11 coefficient for the forward fit and
c11 coefficient for the inverse fit. This allows a direct correlation between the columns and
the coefficients used in the fit with little ambiguity. These columns can then be read into a
polynomial of order NORDER (from the table header). For subarray data, the pixel position from the observation should be corrected for the offset of the subarray to put it into
the coordinate frame of the entire chip.
Figure 1: Schematic of Reference Points for ACS WFC observation.
Chip #1
.a
.C
Chip #2
.b
Once the coefficients have been read in from the table, they need to be applied to the data.
This relies on understanding how to relate the input pixel positions to those expected for
the fit. The model given in the table covers the entire instrument’s field of view, but some
instruments use more than 1 chip for each observation. The fits for each chip are combined
into a global fit with the same common point (xcom, ycom). Figure 1 shows how the common point C relates to each chip in an ACS WFC observation. This common point is
provided in the table in the XCOM,YCOM columns and with the position being given rel-
5
Instrument Science Report ACS 2000-11
ative to each chip’s origin. As seen in this example, the common position can be different
from the reference position for each chip since the reference position corresponds to the
pixel position a target will be placed for an observation. Values for the common point specific to each chip and the reference points for the example in Figure 1 are provided in
Table 1.
Table 1. Table of Reference Positions for Figure 1.
a
2048
1024
Reference position for Chip #1
b
2048
1024
Reference position for Chip #2
C1
2048
-18
Common position used for Chip #1
C2
2048
2065
Common position used for Chip #2
The fit can then be applied to the image pixel positions relative to the reference pixel position for the chip given by (XREF,YREF); namely, by using x = x d – XREF and
y = y d – YREF , where xd and yd are detector pixel positions. Single chip observations
will have XCOM and YCOM at the same position as the reference position XREF,YREF.
In order to combine the corrected chips into a single corrected image, the corrected position for the common position should be calculated relative to each chip’s reference
position using x = XCOM – XREF and y = YCOM – YREF . The final corrected pixel
position from each chip can then be determined by placing it into the combined output
image relative to the corrected common position using X = x c – X com and
Y = y c – X com , where Xcom and Ycom are the corrected common positions.
Ideally, both chips will be perfectly parallel with each other. However, the values of
THETA in the table demonstrate that they are rotated slightly with respect to each other.
The fit for each chip only insures that the distorted and corrected Y axes for a chip remain
at the same angle with respect to the telescope V3 axis. This allows each chip to have different rotations relative to each other. Typically, though, this mis-alignment between the
chips will be less than a degree. This can be corrected for, if desired, by rotating the corrected position for each pixel by half the difference in rotation between the chips in the
observation. This will then align both chips in the output image.
Summary
The geometric distortion for many instruments can be described using the IDCTAB which
contains the coefficients for the SIAF polynomial fit. This table has been defined so that
any instrument whose distortion has been modelled using a polynomial fit can use this
table as a reference file. The table contains information about the reference positions used
for the fit and used for applying that fit to single- and multiple-chip observations. Each
instrument has its own set of unique requirements that may not be fully met by this speci-
6
Instrument Science Report ACS 2000-11
fication. Those requirements can therefore be met with the addition of columns specific to
the instrument. Overall, this table should support a wide-range of instruments and should
be easily applied with software.
7